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The modified Euler method starts with an Euler step, giving a provisional value for $w_{i+1}$ at the next time $t_{i+1}$:

w_{i+1}^\prime = w_i + h f(t_i,w_i).

The step actually taken looks like an Euler step, but with $f$ replaced by the average of $f$ at the starting point of the step and $f$ at the provisional point:
w_{i+1} = w_i + \frac{h}{2}\left[f(t_i,w_i) +
\end{displaymath} (6)

This strategy has the effect of building in the curvature term in the Taylor series expansion of the solution (5), leaving an error of ${\cal O}(h^3)$ per step, so a net error of ${\cal O}[h^2(b-a)]$ in the approximation to $y(b)$ - hence a second order method.

Carleton DeTar 2008-12-01